The Summation Formulae of Euler–Maclaurin, Abel–Plana...

Results. Math. 59 (2011), 359–400c© 2010 Springer Basel AG1422-6383/11/030359-42published online April 2, 2011DOI 10.1007/s00025-010-0083-8 Results in Mathematics

The Summation Formulaeof Euler–Maclaurin, Abel–Plana,Poisson, and their Interconnectionswith the Approximate SamplingFormula of Signal Analysis

P. L. Butzer, P. J. S. G. Ferreira, G. Schmeisser and R. L. Stens

In Memory of Alexander Ostrowski, 1893–1986Dedicated to Heinrich Wefelscheid to mark his 70th birthday

Abstract. This paper is concerned with the two summation formulae ofEuler–Maclaurin (EMSF) and Abel–Plana (APSF) of numerical analysis,that of Poisson (PSF) of Fourier analysis, and the approximate samplingformula (ASF) of signal analysis. It is shown that these four fundamentalpropositions are all equivalent, in the sense that each is a corollary ofany of the others. For this purpose ten of the twelve possible implicationsare established. Four of these, namely the implications of the groupingAPSF ⇐ ASF ⇒ EMSF ⇔ PSF are shown here for the first time. Theproofs of the others, which are already known and were established bythree of the above authors, have been adapted to the present setting.In this unified exposition the use of powerful methods of proof has beenavoided as far as possible, in order that the implications may stand ina clear light and not be overwhelmed by external factors. Finally, thefour propositions of this paper are brought into connection with four

After this manuscript was accepted for publication the authors learned that special issueswere being prepared to honour Heinrich Wefelscheid on the occasion of his 70th anniversary.Since they valued his unstinted service to mathematics as a whole, thus the founding of thepresent journal and his leading editorship over the years, his publication of the CollectedWorks, in several volumes, of W. Blaschke and E. Landau, they expressed interest that thepresent paper, already dedicated to A. Ostrowski, be included in the special issues.

360 P. L. Butzer et al. Results. Math.

propositions of mathematical analysis for bandlimited functions, includ-ing the Whittaker–Kotel’nikov–Shannon sampling theorem. In conclu-sion, all eight propositions are equivalent to another. Finally, the firstthree summation formulae are interpreted as quadrature formulae.

Mathematics Subject Classification (2010). 65B15, 65D32, 94A20.

Keywords. Euler–Maclaurin summation formula, Abel–Plana summationformula, Poisson summation formula, Whittaker–Kotel’nikov–Shannonsampling theorem, Approximate sampling formula, Bandlimited signals,Quadrature formulae.

1. Introduction

First to the Euler–Maclaurin summation formula of the broad area of numer-ical analysis, a very important one, which has indeed many applications.

Euler–Maclaurin Summation Formula (EMSF)

For n, r ∈ N and f ∈ C(2r)[0, n], we have

n∑

k=0

f(k) =

n∫

0

f(x) dx

+12

[f(0) + f(n)] +r∑

k=1

B2k

(2k)!

[f (2k−1)(n) − f (2k−1)(0)

]

+ (−1)r∞∑

k=1

n∫

0

ei2πkt + e−i2πkt

(2πk)2rf (2r)(t) dt, (1.1)

where B2k are the Bernoulli numbers.Euler discovered this summation formula in connection with the so-called

Basel problem, i.e., with determining ζ(2) in modern terminology. It can befound without proof in [23] (submitted 1732) and with a complete deduc-tion in [24] (submitted 1735), where he used it to calculate ζ(2), ζ(3), ζ(4),and the Euler constant γ; for more details see [26]. The formula was foundindependently by Maclaurin in 1738 [38]; cf. [39] for the differences betweenEuler’s and Maclaurin’s approaches. See also the review of Euler’s life andworks by Gautschi [27], and the overview by Apostol [3]. Of further inter-est is that Ramanujan [4, Chaps. 6, 8] introduced a method of summationbased on EMSF. Candelpergher et al. [17,18] presented a rigorous treatmentof the method together with many applications. This formula is also treatedin Ostrowski’s three-volume treatise on differential and integral calculus [42,vol. II, pp. 287–288], which was very popular in German speaking universitiesin the 1960s.

Vol. 59 (2011) Summation Formulae of Euler–Maclaurin, Abel–Plana etc 361

Whereas the Euler–Maclaurin summation formula is a standard one inmathematics per se, our second formula, that of Abel–Plana1 is not. It reads

Abel–Plana Summation Formula (APSF)

Let f be analytic in {z ∈ C : �z ≥ 0}. Suppose that

limy→∞|f(x± iy)|e−2πy = 0 (1.2)

uniformly in x on every finite interval, and that∞∫

0

|f(x+ iy) − f(x− iy)|e−2πy dy (1.3)

exists for every x ≥ 0 and tends to zero when x → ∞. Then

limN→∞

⎡

⎢⎣N∑

k=0

f(k) −N+1/2∫

0

f(x) dx

⎤

⎥⎦ =12f(0) + i

∞∫

0

f(iy) − f(−iy)e2πy − 1

dy. (1.4)

This formula was obtained by Plana [44] in 1820 and independently by Abel[1, p. 23] in 1823 in the form

∞∑

k=0

f(k) −∞∫

0

f(x) dx =12f(0) + i

∞∫

0


dy. (1.5)

As it was customary at that time, neither author gave hypotheses on f .They proceeded via an integral representation of the Bernoulli numbers. Abel[1, p. 27] also established the following interesting “alternating series version”:

∞∑

k=0

(−1)kf(k) =12f(0) + i

∞∫

0

f(iy) − f(−iy)2 sinhπy

dy.

Although his proof was different, this formula can be obtained by apply-ing (1.5) to the function z �→ 2f(2z) and subtracting the resulting equationfrom (1.5).

1 Giovanni (Antonio Amedea) Plana, born 1781 in Voghera, Lombardy (Italy), died 1864 in

Turin, studied 1800–1803 at the Ecole Polytechnique in Paris under Fourier and Lagrange.On the recommendation of Fourier, Plana in 1803 was appointed to the mathematics chairat the Artillery school of Piedmont in Turin. With Lagrange’s recommendation he was alsoappointed to the chair of astronomy at the University of Turin in 1811. He taught bothastronomy and mathematics at both institutions for the rest of his life. His teaching wasof the highest quality. Plana, one of the major Italian scientists of his time, stood in con-tact with Francesco Carlini, Cauchy and Babbage. He was elected to the Turin Academyof Sciences in 1841 and to the Academie des Sciences in Paris in 1860. Plana worked inastronomy (on movement of the moon), Eulerian integrals, elliptic functions, heat, electro-statics and geodesy. See the MacTutor history of mathematics archive http://www-history.mcs.st-andrews.ac.uk/.

http://www-history.mcs.st-andrews.ac.uk/

http://www-history.mcs.st-andrews.ac.uk/


Some authors, [19, I, p. 258, III, p. 53], [45, Sect. 2], consider (1.5) torequire in addition to our assumptions that

∫∞0f(x) dx exists and f(N) → 0

as N → ∞. However, the last assumption is superfluous. Indeed, by (1.4) theexistence of the integral implies the convergence of the series which entailsthat f(N) → 0 as N → ∞.

APSF, as stated by us, is exactly what Henrici proves by contour inte-gration in his book [30, Sect. 4.9]. However, his resulting theorem [30, Theo-rem 4.9 c] is presented in a somewhat different form.

The Abel–Plana formula, connected with several great names of the past,including Cauchy (1826), Kronecker (1889), Lindelof (1905) (see his account ofthe history until 1904 [37, pp. 68–69]), was almost forgotten until Henrici [30,pp. 270–275] brought it up again in his text-book and applied it. One of themost important recent applications of the APSF is to the vacuum expectationvalues for the physical observables in the Casimir2 effect; see [21,40,45,50].For a proof of the functional equation for the Riemann zeta function using theAbel–Plana formula see [51], and for the interconnections of Plana’s formulawith Ramanujan’s integral formula see [55].

The third formula, which is fundamental in a variety of mathematicalfields, the Poisson summation formula, will be stated in the following form(see e.g. [11, p. 202]):

Poisson Summation Formula (PSF)

Let f ∈ L1(R) ∩AC(R). Then∞∑

k=−∞f(k) =

√2π

∞∑

k=−∞f(2πk), (1.6)

where the series on the left-hand side converges absolutely.Above, f denotes the Fourier transform of f ∈ L1(R); it, as well as the

class AC(R), is defined in Sect. 2.Although this formula is named after Poisson, it was given in a general

form without proof by Gauß in a note written on the interior of the cover pageof a book entitled “Opuscula mathematica 1799–1813”; see [25, p. 88]. Jacobiin his work on the transformation formula for elliptic functions [36, p. 260]attributes it to Poisson. Further remarks concerning the history of the Poissonsummation formula can be found in [13, p. 28], [53, pp. 36 f.].

Finally to the approximate sampling formula (ASF) of signal analysis,actually due already to de la Vallee-Poussin (see [20], pp. 65–156 for a repro-duction of this paper and pp. 421–453 for a commentary by Butzer-Stens),Weiss [54], Brown [6], and Butzer-Splettstoßer [14].

2 The Dutch physicist Hendrik Casimir (1909–2000) was awarded an Honorary Doctorate bythe RWTH Aachen University in 1966, the Laudation having been held by Professor JosefMeixner.


Approximate Sampling Formula (ASF)

For f belonging to the class F 2 (see Sect. 2 for definition), w > 0, we have

f(t) =∞∑

k=−∞f

(k

w

)sinc(wt− k) + (Rwf)(t) (t ∈ R), (1.7)

where the remainder Rwf is defined by

(Rwf)(t) :=1√2π

∞∑

k=−∞

(1 − e−i2πkwt

)(2k+1)πw∫

(2k−1)πw

f(v)eivt dv (t ∈ R),

(1.8)

the sinc-function being given by

sinc z :=

{ sin πzπz if z ∈ C\{0},1 if z = 0.

It is the sampling theorem for not necessarily bandlimited functions, general-izing the classical Whittaker–Kotel’nikov–Shannon sampling theorem (CSF)of signal analysis

f(t) =∞∑

k=−∞f

(k

w

)sinc(wt− k) (t ∈ R) (1.9)

for f ∈ B2πw (for definition see Sect. 2). To proceed from (1.9) to (1.7) one has

to add the “error” term (Rwf)(t).

Remark. A consequence of the definition (1.8) of Rwf is the inequality

∣∣(Rwf)(t)∣∣ ≤√

2π

∫

∣∣v∣∣≥πw

|f(v)| dv, (1.10)

yielding

limw→∞

∞∑

k=−∞f

(k

w

)sinc(wt− k) = f(t)

uniformly for t ∈ R.

The implications to be established in this paper between the assertionsof the summation formulae of Euler–Maclaurin (EMSF), Abel–Plana (APSF),Poisson (PSF) and the approximate sampling formula (ASF), are indicated bythe arrows of the graphic in Fig. 1. As the reader observes, there are 12 possibleimplications; of these we shall prove 10 implications, namely the grouping

APSF ⇐= ASF =⇒ EMSF ⇐⇒ PSF


APSFEMSF

ASF

PSF

Figure 1. Diagram of the implications to be proven

as well as the grouping

ASF ⇐⇒ PSF =⇒ APSF ⇐⇒ EMSF =⇒ ASF.

The conclusion: All the formulae are equivalent to each other.The first grouping consists of four new implications which are established

in this paper for the first time. Of the second grouping, the equivalence ASF ⇔PSF, was first presented in the proceedings of the Edmonton conference of 1982[15], the implication EMSF ⇒ ASF in the special issues dedicated to the 90thbirthday of A. Ostrowski in 1983 [16], and that of PSF ⇒ APSF ⇔ EMSF inmemory of U. N. Singh in 1994 [49].

The proofs of five of these six known implications of the second group-ing, established by three of the present authors since 1982, are to be foundin conference proceedings and special issues not always readily available. Forthis reason they are presented here with complete proofs, especially to makesure that they do not involve “circular arguments”, so important in establish-ing equivalences between many assertions. Further, they have been adaptedto those of the new implications of the first grouping in order to give a unifiedexposition of the material. So the reader will find full proofs of all of the resultsdiscussed in the present paper.

All the proofs use only a handful of modest side results, namely classi-cal Fourier analysis, the Weierstraß theorem for algebraic (not trigonometric)polynomials and Cauchy’s theorem of complex function theory. Therefore, ifwe prove an implication X ⇒ Y, then it is fully justified to say that Y is adirect corollary of X. Here “direct” means that the proof of the implicationproceeds directly from X to Y, and is not of the form X ⇒ Z ⇒ Y, where Z isany other of the formulae under consideration.

The implications APSF ⇒ ASF and APSF ⇒ PSF are left open. Theproblem is that APSF requires functions which are analytic in a half-plane,whereas ASF and PSF hold for larger function classes. Thus it seems to be


appropriate to proceed via a density argument based on an approximation pro-cess which could be so cumbersome that one could not speak of “establishinga direct corollary”.

For those readers who are primarily interested in the equivalence of thefour formulae, the presentation also allows several closed loops of implications,one being PSF ⇒ APSF ⇒ EMSF ⇒ ASF ⇒ PSF. In this case only the proofsof four implications are needed.

Concerning contents, Sect. 2 is devoted to notations and side resultsneeded. Section 3 is concerned with the proof of the equivalence of EMSF andAPSF, and Sect. 4 with the equivalence of PSF and ASF. Section 5 deals withthe PSF ⇔ EMSF and Sect. 6 with EMSF ⇔ ASF. Section 7 treats the APSFas a corollary of ASF and PSF.

Section 8 presents a proof of EMSF; thus since one of the formulae hasbeen shown to be actually valid, the four formulae are not only equivalentamong themselves, but are all valid too.

Section 9 is concerned with APSF, EMSF and PSF considered as quad-rature formulae in case the functions in question are bandlimited. Finally,Sect. 10 compares the results presented here with a similar equivalence group-ing for bandlimited functions in [7]. Section 11 contains a short biography ofA. Ostrowski.

2. Notations and Side Results

In what follows, the Fourier transform of f ∈ Lp(R), p = 1, 2, is defined by

f(v) :=1√2π

∫

R

f(u)e−ivu du (v ∈ R),

the integral being understood as the limit in the L2(R)-norm for p = 2. Weshall consider the following function spaces:

F p :={f : R → C; f ∈ Lp(R) ∩ C(R), f ∈ L1(R)

}(p = 1, 2),

where C(R) denotes the space of all uniformly continuous and bounded func-tions on R. Since f ∈ L1(R) implies that f is bounded, there holds F 1 ⊂ F 2.

We also make use of the space AC(R) = AC(1)(R). It comprises all func-tions f : R → C being absolutely continuous on R, i.e., those f having therepresentation

f(t) =

t∫

−∞f ′(u) du+ c (t ∈ R)

with f ′ ∈ L1(R). More generally, AC(r+1)(R), r ∈ N, consists of all functionsf with f ′ ∈ AC(r)(R) ∩ L1(R).


The Bernstein spaces Bpσ for p = 1, 2 and σ > 0 are defined in terms of

the Fourier transform via

Bpσ :=

{f ∈ Lp(R); f(v) = 0 a.e. outside [−πσ, πσ]

}.

In view of the Paley–Wiener theorem (see e.g. [56, vol. II, p. 272]) these spacescan be equivalently characterized as the set of all entire functions of exponen-tial type σ, the restriction to R of which belongs to Lp(R). For this reason thespaces B2

σ are also known as Paley–Wiener spaces.Let us list some well-known results from Fourier analysis; see any text-

book in the matter, e.g. [11, Sects. 5.1, 5.2].

Proposition 2.1. a) If f ∈ Lp(R), p = 1, 2, then for each h ∈ R,

[e−ih·f(·)]∧(v) = f(v + h) (2.1)

for all v ∈ R in case p = 1 and a. e. in case p = 2.b) If f ∈ L1(R), g ∈ Lp(R), p = 1, 2, then the convolution

(f ∗ g)(t) :=1√2π

∫

R

f(u)g(t− u) du (2.2)

belongs to Lp(R), and for the Fourier transform one has the convolutiontheorem

(f ∗ g)∧(v) = f(v) g(v) (2.3)

for all v ∈ R in case p = 1 and a. e. in case p = 2.c) If f, g ∈ Lp(R), p = 1, 2, then there holds the exchange formula

∫

R

f(v)g(v) dv =∫

R

f(v)g(v) dv. (2.4)

d) If f ∈ L2(R), then f also belongs to L2(R) and ‖f‖2 = ‖f‖2. Further-more, there holds the inversion formula

f(t) =1√2π

∫

R

f(v)eivt dv = f(−t) a. e., (2.5)

where the integral is again understood as the limit in L2(R)-norm. Iff ∈ F p, p = 1, 2, then the integral in (2.5) exists as an ordinary Lebesgueintegral and both equalities hold for all t ∈ R.

Lemma 2.2. There hold the formulae[

1√2π w

rect( ·w

)ei·t]∧

(v) = sincw(v − t) = sincw(t− v) (t ∈ R)

(2.6)

[sincw(· − t)]∧ (v) =1√

2π wrect

( vw

)e−ivt (t ∈ R) (2.7)


with the rectangle function

rect(t) :=

⎧⎪⎨

⎪⎩

1, |t| < π12 , |t| = π

0, |t| > π

.

Proof. Equation (2.6) is shown by a simple integration, and (2.7) follows from(2.6) by the Fourier inversion formula. (For an elementary proof of (2.7) with-out using the inversion formula see [5, p. 13 f].) �

Proposition 2.1 and Lemma 2.2 enable us to prove alternative represen-tations of the sampling series

(Swf)(t) :=∞∑

k=−∞f

(k

w

)sincw(t− k) (2.8)

and the remainder Rwf of (1.8).

Proposition 2.3. For fixed t ∈ R, w > 0 let et,w(v), v ∈ R, be the functionobtained by restricting v �→ eitv to the interval [−πw, πw), and then extendingit to R by 2πw-periodic continuation. Then for f ∈ F 2,

(Swf)(t) =1√2π

∫

R

f(v)et,w(v) dv. (2.9)

Proof. By a simple calculation et,w has the Fourier expansion

et,w(v) =∞∑

k=−∞sinc(wt− k)eikv/w. (2.10)

Since et,w is of bounded variation, its Fourier series converges and (2.10) holdsat each point of continuity. Moreover, its partial sums

sn(v) :=∑

|k|≤n

sincw(t− k)eikv/w (2.11)

are uniformly bounded with respect to v ∈ R and n ∈ N, that is

|sn(v)| ≤ C (v ∈ R;n ∈ N)

for some constant C > 0 depending on t only; see [56, p. 90, Theorem 3.7].Since f ∈ L1(R), we see that f · sn has an absolutely integrable majorant forall n ∈ N. Therefore, Lebesgue’s dominated convergence theorem allows us to


conclude that1√2π

∫

R

f(v)et,w(v) dv = limn→∞

1√2π

∫

R

f(v)sn(v) dv

= limn→∞

∑

|k|≤n

⎧⎨

⎩1√2π

∫

R

f(v)eikv/w dv

⎫⎬

⎭ sinc(wt− k)

= limn→∞

∑

|k|≤n

f

(k

w

)sincw(t− k),

where the Fourier inversion theorem (Proposition 2.1 d) has been used in thelast step. �Proposition 2.4. For f ∈ F 2 and w > 0 we have the following two alternativerepresentations of the remainder Rwf of (1.8),

(Rwf)(t) =∞∑

k=−∞

(ei2πkwt − 1

) ∫

R

f(u)e−i2πkwu sincw(t− u) du (2.12)

= f(t) −∞∑

k=−∞

∫

R

f(u)e−i2πkwu sincw(t− u) du (t ∈ R).

(2.13)

Proof. Using (2.4), (2.6) and (2.1) we obtain(2k+1)πw∫

(2k−1)πw

f(v)eitv dv = ei2πkwt

∫

R

f(v + 2πkw) rect( vw

)eitv dv

=√

2π ei2πkwt

∫

R

f(u)e−i2πkwu sinc(t− u) du.

(2.14)

Inserting this into (1.8) yields (2.12).Moreover, (1.8) can be rewritten as

(Rwf)(t) =1√2π

∞∫

−∞f(v)eivt dv

− 1√2π

∞∑

k=−∞e−i2πkwt

(2k+1)πw∫

(2k−1)πw

f(v)eivt dv.

Noting that the first integral equals f(t) by the Fourier inversion formula(2.5), and inserting (2.14) into the infinite series, we obtain the representation(2.13). �


We also need the following variant of Nikolskiı’s inequality. Its proof isnearly the same as that of [41, Theorem 3.3.1]. It requires only the mean valuetheorems of integral calculus as well as Minkowski’s inequality for sums andHolder’s inequality for integrals.

Lemma 2.5. For f ∈ L1(R) ∩AC(R) there holds∞∑

k=−∞|f(k)| ≤ ‖f‖L1(R) + ‖f ′‖L1(R). (2.15)

In the proofs below it will often be convenient to prove the result first fora particular case, and then the general case by a density argument using a suit-able approximation. In this respect we will make use of the singular integralof Fejer or Fejer means (cf. (2.2)), namely,

(σρf)(t) := (f ∗ χρ)(t) (t ∈ R, ρ > 0), (2.16)

χρ being Fejer’s kernel

χρ(u) := ρ√

2π sinc2(ρu) (u ∈ R, ρ > 0), (2.17)

having Fourier transform

χρ(v) =(

1 − |v|2πρ

)

+

(v ∈ R, ρ > 0), (2.18)

where h+(u) := h(u) if h(u) ≥ 0, and h+(u) = 0 if h(u) < 0.

Proposition 2.6. a) If f ∈ Lp(R), 1 ≤ p < ∞, then for each r ∈ N,

σρf ∈ Lp(R) ∩ C(r)(R) with (σρf)(r) ∈ Lp(R) (ρ > 0)

and

limρ→∞‖σρf − f‖Lp(R) = 0.

Furthermore, there holds limρ→∞(σρf)(t) = f(t) at each point t ∈ R

where f is continuous.b) If f ∈ Lp(R) ∩C(s−1)(R) with f (s) ∈ Lp(R) for some s ∈ N and 1 ≤ p <

∞, then

limρ→∞‖(σρf)(s) − f (s)‖Lp(R) = 0.

c) If f ∈ F 2, then for each w > 0,

limρ→∞(Sw(σρf))(t) = (Swf)(t) (t ∈ R), (2.19)

and

limρ→∞(Rw(σρf))(t) = (Rwf)(t) (t ∈ R). (2.20)


d) If f ∈ L1(R) ∩AC(R), then

limρ→∞

∞∑

k=−∞(σρf)(k) =

∞∑

k=−∞f(k), (2.21)

all series being absolutely convergent.

Proof. Parts a) and b) are well known (see e.g. [11, pp. 122, 134]). As to c), onehas by the convolution theorem (2.3) and by (2.9), noting that |et,w(v)| = 1,

∣∣(Sw(σρf))(t) − (Swf)(t)∣∣ ≤ 1√

2π

∞∫

−∞

∣∣f(v)∣∣∣∣∣∣

(1 − |v|

2πρ

)

+

− 1∣∣∣∣dv.

Since f ∈ L1(R), the last integral tends to zero for ρ → ∞ by Lebesgue’sdominated convergence theorem. This yields (2.19).

Similarly, one has by (1.8),

∣∣(Rw(σρf))(t) − (Rwf)(t)∣∣ ≤√

2π

∞∑

k=−∞

(2k+1)πw∫

(2k−1)πw

|f(v)|∣∣∣∣

(1 − |v|

2πρ

)

+

− 1∣∣∣∣dv

=

√2π

∞∫

−∞

∣∣f(v)∣∣∣∣∣∣

(1 − |v|

2πρ

)

+

− 1∣∣∣∣dv,

which proves (2.20).d) The proof of the absolute convergence of the series follows by

Lemma 2.5, noting that the assumptions upon f imply that σρf ∈ L1(R) ∩AC(R) by part a). Assertion (2.21) is now an easy consequence of Lemma 2.5and part b), since

∞∑

k=−∞

∣∣(σρf)(k) − f(k)∣∣ ≤ ‖σρf − f‖L1(R) + ‖(σρf

′) − f ′‖L1(R).

�

Lemma 2.7. a) Let g be a continuous λ-periodic function with∫ λ

0g(u)du =

0. Then

∣∣∣ρ∫

1

g(u)u

du∣∣∣ ≤ 2

λ∫

0

|g(u)| du

for ρ ∈ [1,∞).b) There exists a constant C ∈ R, such that for all a, b, t ∈ R and k ∈ Z,

∣∣∣b∫

a

sinc(t− u)ei2πku du∣∣∣ ≤ C.


Proof. By the second law of the mean for integrals, there exists ξ ∈ (1, ρ) suchthat

ρ∫

1

g(u)u

du =

ξ∫

1

g(u) du+1ρ

ρ∫

ξ

g(u) du.

Thus∣∣∣

ρ∫

1

g(u)u

du∣∣∣ ≤(

1 +1ρ

) λ∫

0

|g(u)| du ≤ 2

λ∫

0

|g(u)| du.

Part b) follows immediately by choosing g(u) := sinπ(t− u)ei2πu in a). �

3. The Summation Formulae of Euler–Maclaurinand Abel–Plana

The summation formulae of Euler–Maclaurin and Abel–Plana can be inter-preted as trapezoidal rules with remainders for quadrature over [0, n] and[0,∞], respectively; see Sect. 9 below. As such, one may suspect that APSF issomehow the limiting case n → ∞ of EMSF. However, the situation is moreintricate since the hypotheses on f and the representations of the remain-ders in the two formulae are very different. Nevertheless, it was shown in [49]that EMSF and APSF can be deduced from each other.3 The cited paperestablished a conjecture raised by one of the authors at a lecture held at theUniversity of Erlangen-Nuremberg in February 1991.

The basic idea is to transform the remainders by using Cauchy’s theo-rem for contour integration of analytic functions along rectangles. However,in the hypotheses of EMSF the function f need not be analytic. Therefore fis approximated by an analytic function ϕ so that APSF becomes applicable.Such a function ϕ is obtained by a process based on the classical approximationtheorem of Weierstraß.

Since the paper [49] is not easily available, we borrow details from it andpresent them here in a somewhat different form.

For our purpose it is convenient to introduce

Lj(z) := (−1)j∞∑

k=1

e−2πkz

(2πk)j(j ∈ N0). (3.1)

For j = 0 and j = 1 the series converges in the open right half-plane, where itdefines Lj as an analytic function.

Using the formula for the geometric series, we obtain

L0(z) =1

e2πz − 1(3.2)

3 Hardy [29, Sect. 13.14] only indicated a proof for EMSF ⇒ APSF, but he needed additionalassumptions on f for carrying it out.


and may therefore consider L0 a meromorphic function which has simple polesat z ∈ iZ and is analytic in C\iZ.

In the case j = 1 the Cauchy-Schwarz inequality implies for x > 0 andy ∈ R that

|L1(x+ iy)| ≤( ∞∑

k=1

1(2πk)2

)1/2( ∞∑

k=1

e−4πkx

)1/2

=1

2√

6

(e4πx − 1

)−1/2,

and so

L1(x+ iy) = O(x−1/2) (x → 0+). (3.3)

For j ≥ 2 the functions Lj are defined in the closed right half-plane andanalytic in its interior. Two useful obvious properties are

Lj(z + i) = Lj(z), L′j+1(z) = Lj(z) (3.4)

holding for �z > 0 and all j ∈ N0. We also mention thatB2k

(2k)!= (−1)k+12L2k(0). (3.5)

As in [49], we will consider EMSF for r = 1 only, since the more generalform (1.1) can be deduced from that special case by repeated integration byparts of the remainder. Employing the functions Lj , we may write EMSF, thus(1.1), for r = 1, as

n∑

k=0

f(k) =12[f(0) + f(n)

]+

n∫

0

f(x) dx

+112[f ′(n) − f ′(0)

]−n∫

0

[L2(it) + L2(−it)

]f ′′(t) dt (3.6)

while APSF (1.4) takes the form

limN→∞

{ N∑

k=0

f(k) −N+1/2∫

0

f(x) dx}

=12f(0) + i

∞∫

0

L0(y) [f(iy) − f(−iy)] dy. (3.7)

The crucial link between the two summation formulae (3.6) and (3.7) iscontained in the following lemma.

Lemma 3.1. Let f satisfy the hypotheses of APSF. Then the expressions

An(f) := i

∞∫

0

L0(t) [f(it) − f(−it)] dt− i

∞∫

0

L0(t) [f(n+ it) − f(n− it)] dt


and

Bn(f) :=112[f ′(n) − f ′(0)

]−n∫

0

[L2(it) + L2(−it)

]f ′′(t) dt

are equal.

Proof. For z0, z1 ∈ C, we mean by∫ z1

z0. . . dz the integral along the line segment

starting at z0 and ending at z1.Let ε > 0 and K > ε. Introducing

F (z) := L0(−iz)f(z) and G(z) := L0(iz)f(z),

we define

I+n (ε,K) :=

iK∫

iε

F (z) dz −n+iK∫

n+iε

F (z) dz

and

I−n (ε,K) :=

−iK∫

−iε

G(z) dz −n−iK∫

n−iε

G(z) dz.

Then the expression An(f) can be rewritten as

An(f) = limε→0+

limK→+∞

[I+n (ε,K) + I−

n (ε,K)]. (3.8)

Clearly, F is analytic in {z ∈ C : �z ≥ 0, � z ≥ ε}. Hence, by Cauchy’stheorem, integration of the function F along the rectangle with vertices atiε, n+ iε, n+ iK, iK yields

I+n (ε,K) =

n+iε∫

iε

F (z) dz −n+iK∫

iK

F (z) dz.

As a consequence of (1.2), the last integral vanishes as K → ∞. Analogousconsiderations hold for G in {z ∈ C : � z ≥ 0, � z ≤ −ε}, and so (3.8) reducesto

An(f) = limε→0+

⎡

⎣n+iε∫

iε

F (z) dz +

n−iε∫

−iε

G(z) dz

⎤

⎦. (3.9)

Now integrating by parts twice and using (3.4), we find thatn+iε∫

iε

F (z) dz =

n∫

0

L0(ε− it)f(t+ iε) dt

= [iL1(ε)f(t+ iε) + L2(ε)f ′(t+ iε)]t=nt=0 −

n∫

0

L2(ε− it)f ′′(t+iε) dt. (3.10)


Similarly for the second integral in (3.9),n−iε∫

−iε

G(z) dz =

n∫

0

L0(ε+ it)f(t− iε) dt

= [−iL1(ε)f(t−iε)+L2(ε)f ′(t−iε)]t=nt=0 −

n∫

0

L2(ε+it)f ′′(t−iε) dt. (3.11)

But from (3.3) it follows that

L1(ε)[f(t+ iε) − f(t− iε)

]= O

(ε1/2)

→ 0 (ε → 0+).

Hence, combining (3.9)–(3.11) and letting ε → 0+, we obtain, since 2L2(0) =B2/2 = 1/12,

An(f) = 2L2(0)[f ′(n) − f ′(0)

]−n∫

0

[L2(−it) + L2(it)

]f ′′(t) dt = Bn(f)

as was to be shown. �

Now we turn to the aforementioned approximation process.

Lemma 3.2. Let f ∈ C(2)[0, n], where n ∈ N, and denote by ‖ · ‖C[0,n] thesupremum norm on [0, n]. Then, given ε > 0, there exists a function ϕsatisfying the hypotheses of APSF such that

‖ f − ϕ ‖C[0,n] ≤ nε

2 (n2 + 1)(3.12)

and

‖ f ′′ − ϕ′′ ‖C[0,n] ≤ ε

n (n2 + 1). (3.13)

Proof. By the approximation theorem of Weierstraß, there exists a polynomialP (x) such that

‖ f ′′ − P ‖C[0,n] ≤ ε

2n(n2 + 1). (3.14)

Defining

Q(x) := f(0) + xf ′(0) +

x∫

0

(x− t)P (t) dt,

and choosing λ ∈ (0, 2π) such that

λ(n ‖Q′′ ‖C[0,n] + 2 ‖Q′ ‖C[0,n] + 2π ‖Q ‖C[0,n]

)≤ ε

2n (n2 + 1), (3.15)

we claim that ϕ(z) := e−λzQ(z) has all the desired properties.


Indeed, since Q is a polynomial and λ > 0, it is easily verified that ϕsatisfies the hypotheses of APSF.

Next, writing f as

f(x) = f(0) + xf ′(0) +

x∫

0

(x− t)f ′′(t) dt,

which is a special case of Taylor’s formula, and recalling the definition of Q,we find that

|f(x) −Q(x)| ≤∣∣∣∣

x∫

0

(x− t) (f ′′(t) − P (t)) dt∣∣∣∣ ≤

n2

2‖ f ′′ − P ‖C[0,n]

(3.16)

for x ∈ [0, n]. On the same interval,

|1 − e−λx| ≤ 1 − e−λn ≤ nλ. (3.17)

Since

f(x) − ϕ(x) = f(x) −Q(x) +(1 − e−λx

)Q(x),

we may employ (3.16) and (3.17) to deduce that

‖ f − ϕ ‖C[0,n] ≤ n2

2‖ f ′′ − P ‖C[0,n] + nλ ‖Q ‖C[0,n] .

Now, estimating the right-hand side with the help of (3.14) and (3.15), we seethat (3.12) holds.

Finally, since Q′′(x) ≡ P (x), we have

f ′′(x) − ϕ′′(x) = f ′′(x) − P (x) + Q′′(x) − d2

dx2

(e−λxQ(x)

)

and deduce that

‖ f ′′ − ϕ′′ ‖C[0,n] ≤ ‖ f ′′ − P ‖C[0,n] + nλ ‖Q′′ ‖C[0,n]

+ 2λ ‖Q′ ‖C[0,n] + λ2 ‖Q ‖C[0,n] .

In conjunction with (3.14) and (3.15), this implies (3.13). �

Proof of APSF ⇒ EMSF. As mentioned above, it suffices to prove (3.6). If fsatisfies the hypotheses of APSF, then so does f(· + n). Applying APSF tothese two functions and subtracting the results, we obtain

n−1∑

k=0

f(k) =12[f(0) − f(n)

]+

n∫

0

f(x) dx+An(f)

with An(f) as defined in Lemma 3.1. The conclusion of Lemma 3.1 gives (3.6)immediately.


Now suppose that f satisfies the hypotheses of EMSF only, i.e. f ∈C(2)[0, n]. We have to show that

E(f) :=12

[f(0) + f(n)] +n−1∑

k=1

f(k) −n∫

0

f(x) dx− 112[f ′(n) − f ′(0)

]

−n∫

0

[L2(it) − L2(−it)

]f ′′(t) dt

vanishes. Given any ε > 0, we now employ Lemma 3.2 and define ψ := f − ϕ.By what we have proved already, namely that E(ϕ) = 0,

E(f) = E(ϕ) + E(ψ) = E(ψ).

Finally, noting that

∣∣L2(it) − L2(−it)∣∣ ≤∣∣∣∣

∞∑

k=1

2 cos(2πkt)(2πk)2

∣∣∣∣ ≤1

2π2

∞∑

k=1

1k2

=112

for t ∈ R, and using (3.12) and (3.13), we conclude that

|E(ψ)| ≤ ‖ψ ‖C[0,n] + (n− 1) ‖ψ ‖C[0,n] + n ‖ψ ‖C[0,n]

+112

n∫

0

|ψ′′(t)| dt+n

12‖ψ′′ ‖C[0,n]

≤ 2n ‖ψ ‖C[0,n] +n

6‖ψ′′ ‖C[0,n] < ε.

This implies that E(f) = 0. �

Proof of EMSF ⇒ APSF. Suppose that f satisfies the hypotheses of APSFand that (3.6) holds. Employing Lemma 3.1, we may rewrite (3.6) as

n∑

k=0

f(k) =12

[f(0) + f(n)] +

n∫

0

f(x) dx

+i

∞∫

0

L0(t) [f(it) − f(−it)] dt

−i∞∫

0

L0(t) [f(n+ it) − f(n− it)] dt. (3.18)

Now let ε > 0 and K > ε. Consider the rectangle with vertices at

n+ iε, n+12

+ iε, n+12

+ iK, n+ iK


and its conjugate in the lower half-plane. By contour integration of the functionL0(−iz)f(z) along the first rectangle and L0(iz)f(z) along the second, we find,as in the proof of Lemma 3.1, that

i

∞∫

0

L0(t) [f(n+ it) − f(n− it)]dt

− i

∞∫

0

L0

(t+

12i

)[f

(n+

12

+ it

)− f

(n+

12

− it

)]dt

= limε→0+

n+1/2∫

n

[L0(ε− it)f(t+ iε) + L0(ε+ it)f(t− iε)

]dt =: Cn(f).

Next we observe that

limε→0+

n+1/2∫

n

L0(ε∓ it) [f(t± iε) − f(t)] dt

= limε→0+

n+1/2∫

n

ε

e2π(ε∓it) − 1· f(t± iε) − f(t)

εdt = 0

since the integrand remains bounded on [n, n+ 12 ] as ε → 0+ and approaches

zero on (n, n+ 12 ]. Hence Cn(f) can be expressed as

Cn(f) = limε→0+

n+1/2∫

n

[L0(ε− it) + L0(ε+ it)] f(t) dt

= limε→0+

n+1/2∫

n

2∞∑

k=1

e−2πkε cos(2πkt)f(t) dt

= limε→0+

∞∑

k=1

e−2πkεak,

where

ak := 2

n+1/2∫

n

f(t) cos(2πkt) dt.


We now interpret ak as the coefficients of a Fourier series. Indeed, let g be the1-periodic continuation of the function

g(x) :=12f(n+ |x− n|) −

n+1/2∫

n

f(t) dt

defined on [n− 12 , n+ 1

2 ]. Then g is continuous and its associated Fourier seriesis

g(x) ∼∞∑

k=1

ak cos(2πkx).

Hence Cn(f) is the Abel–Poisson limit of the Fourier series of g at x = 0; see[11, p. 46, Proposition 1.2.8]. Therefore,

Cn(f) = g(0) = g(n) =12f(n) −

n+1/2∫

n

f(t) dt.

Altogether, we can rewrite (3.18) as

n∑

k=0

f(k) =12f(0)

n+1/2∫

0

f(x) dx+ i

∞∫

0

L0(t)[f(it) − f(−it)]dt

− i

∞∫

0

L0

(t+

12i

)[f

(n+

12

+ it

)y − f

(n+

12

− it

)]dt.

Since

L0

(t+

i

2

)= − 1

e2πt + 1,

we see from (1.3) that the last integral vanishes as n → ∞. Thus we arriveat (3.7). �

4. Poisson’s Summation Formula and the ApproximateSampling Formula

When we want to deduce ASF from one of the other formulae, it suffices to doit for w = 1. Indeed, the general formula is obtained from that special case byreplacing f by f(·/w), t by wt and noting that

[(R1f)

( ·w

)](wt) = (Rwf)(t).

The two proofs of this section are adapted from [15].


Proof of PSF ⇒ ASF. First we assume that f ∈ L2(R) ∩ AC(R). Then, forfixed t ∈ R, the function gt(·) := f(·) sinc(t − ·) belongs to L1(R) ∩ AC(R),and PSF (1.6) applies. Noting that

∞∑

k=−∞gt(k) = (S1f)(t)

and√

2π gt(2πk) =∫

R

f(u)e−i2πku sinc(t− u) du,

PSF yields

(S1f)(t) =∞∑

k=−∞

∫

R

f(u)e−i2πku sinc(t− u) du = f(t) − (R1f)(t),

the representation (2.13) being used for the remainder R1f . This is ASF forw = 1 in that particular case.

In order to prove ASF for f ∈ F 2, we approximate f by the convolutionintegral of Fejer σρf ; see (2.16). According to Proposition 2.6 a), σρf satisfiesthe assumptions of the particular case, hence

(S1(σρf))(t) = (σρf)(t) − (R1(σρf))(t) (t ∈ R, ρ > 0)). (4.1)

Letting ρ → ∞, ASF for w = 1 follows again by Proposition 2.6. �

Next to the inverse implication.

Proof of ASF ⇒ PSF. Assume that f ∈ L1(R)∩AC(R) with f ∈ L1(R). ThenASF applies to f and we obtain for w = 1,

f(t) =∞∑

k=−∞f(k) sinc(t− k)

+∞∑

k=−∞

(ei2πkt − 1

) ∫

R

f(u)e−i2πku sinc(t− u) du (t ∈ R), (4.2)

where we have used the remainder in the form (2.12). Integrating this equationterm by term formally, one obtains

∫

R

f(t) dt =∞∑

k=−∞f(k)

∫

R

sinc(t− k) dt

+∞∑

k=−∞

∫

R

f(u)e−i2πku

{∫

R

(ei2πkt − 1

)sinc(t− u) dt

}du.


Now, the integrals in the first series are equal to 1 by (2.7), and, similarly, theintegrals in curly brackets are equal to −1 for k �= 0 and equal to 0 for k = 0,again by (2.7). So one has

∫

R

f(t) dt =∞∑

k=−∞f(k) −

√2π

∞∑

k=−∞k =0

f(2πk),

which is PSF.To make these considerations rigorous, first assume that f(v) = 0 for

|v| > ρ for some ρ > 0. This implies in particular that f ∈ L1(R). Now itfollows from (2.14) that the integrals in (4.2) vanish for |k| > ρ, i.e., (4.2) canbe rewritten as

f(t) =∞∑

k=−∞f(k) sinc(t− k)

+∑

|k|≤ρ

(ei2πkt − 1

) ∫

R

f(u)e−i2πku sinc(t− u) du (t ∈ R). (4.3)

Since the infinite series is uniformly convergent with respect to t in viewof (2.15), one can integrate (4.3) term by term from −R to R to deduce

R∫

−R

f(t) dt =∞∑

k=−∞f(k)

R∫

−R

sinc(t− k) dt

+∑

|k|≤ρ

∫

R

f(u)e−i2πku

{ R∫

−R

(ei2πkt − 1

)sinc(t− u) dt

}du, (4.4)

where the interchange of the order of integration is justified by Fubini’stheorem.

Now take the limit for R → ∞. As to the infinite series, since the integralsare uniformly bounded with respect to R > 0 and k ∈ Z by Lemma 2.7 b), onecan interchange the limit with summation. Similarly, in the second term onthe right-hand side of (4.4), the inner integrals are uniformly bounded withrespect to R > 0 and u ∈ R by Lemma 2.7 b), and hence one can also inter-change the limit for R → ∞ with the outer integral. So one obtains from (4.4)for R → ∞,

∞∫

−∞f(t) dt =

∞∑

k=−∞f(k)

∞∫

−∞sinc(t− k) dt

+∑

|k|≤ρ

∫

R

f(u)e−i2πku

{ ∞∫

−∞

(ei2πkt − 1

)sinc(t− u) dt

}du.


This yields PSF as in the formal proof, noting that by assumption f(2πk) = 0for |k| > ρ.

It remains to show that the additional assumption f(v) = 0 for |v| > ρcan be dropped. To this end, we consider the Fejer means σρf (cf. (2.16)). Itfollows from (2.3), (2.18) and Proposition 2.6 a) that σρf satisfies the assump-tions of the particular case, and hence

∞∑

k=−∞(σρf)(k) =

√2π∑

|k|≤ρ

f(2πk)(

1 − |k|ρ

).

Letting now ρ → ∞ we obtain by Proposition 2.6 d) that∞∑

k=−∞f(k) = lim

ρ→∞√

2π∑

|k|≤ρ

f(2πk)(

1 − |k|ρ

). (4.5)

Finally, since f ∈ L1(R) ∩ AC(R) implies that f(v) = o(|v|−1

)for |v| → ∞

(cf. [11, p. 194]), it follows that

limρ→∞

∑

|k|≤ρ

f(2πk)|k|ρ

= 0,

so that one can replace the right-hand side of (4.5) by√

2π∑∞

−∞ f(2πk),giving PSF. �

5. Poisson’s Summation Formula and the Euler–MaclaurinFormula

Proof of PSF ⇒ EMSF. Let f satisfy the hypothesis of EMSF for r = 1, thatis, f ∈ C(2)[0, n]. We may and will assume that f(0) = f(n) = 0. Indeed, wesimply have to add to f an appropriate linear function and observe that EMSFis trivial for linear functions.

Defining

φ(x) :=

{f(x), if x ∈ [0, n]

0, if x ∈ R\(0, n),(5.1)

we see that

φ(v) =1√2π

n∫

0

f(u)e−iuv du.

Integrating by parts twice, we find that

φ(v) =1√

2πv2

⎡

⎣f ′(n)e−inv − f ′(0) −n∫

0

f ′′(u)e−iuv du

⎤

⎦ (5.2)


for v �= 0. Hence there exist constants c1 > 0 and c2 > 0 such that

|φ(v)| ≤ min{c1,

c2v2

}(v ∈ R). (5.3)

With this it is readily seen that φ satisfies the hypothesis of Poisson’s summa-tion formula. Hence we obtain

n∑

k=0

f(k) =√

2π∞∑

k=−∞φ(2πk). (5.4)

Finally, calculating φ(2πk) for k �= 0 with the help of (5.2) and using that∞∑

k=1

2(2πk)2

=B2

2!=

112, (5.5)

we arrive atn∑

k=0

f(k) =

n∫

0

f(t) dt+112[f ′(n) − f ′(0)

]

−∞∑

k=1

n∫

0

ei2πku + e−i2πku

(2πk)2f ′′(u) du.

This is EMSF for r = 1 and functions satisfying f(0) = f(n) = 0. �

The implication PSF ⇒ EMSF is new. A variant with n = ∞ forfunctions belonging to the Schwarz space S of rapidly decreasing functionscan be found in [22, p. 112 ff.]. It is also proved as a corollary of PSF.

Now to the converse implication which is also new.

Proof of EMSF ⇒ PSF. Starting with EMSF for r = 1, we find for f ∈C(2)[0, n] by an integration by parts of (1.1) that

n∑

k=0

f(k) =

n∫

0

f(x) dx+12

[f(0) + f(n)] − 1π

n∫

0

∞∑

k=1

sin(2πkt)k

f ′(t) dt. (5.6)

It is known and can be shown in an elementary way that4

∞∑

k=1

sin(2πkt)k

= −π(t− �t� − 1

2

)(5.7)

for t ∈ R\Z. By Abelian summation and standard estimates, it can also beshown that the partial sums of the series are uniformly bounded on the wholeof R.

Although (5.6) was derived for f ∈ C(2)[0, n], it is valid for functionsthat are absolutely continuous on [0, n]. This follows by a density argument

4 This result can be found in many textbooks on Analysis for first year students.


and a limiting process which is easy since in (5.6) we have a finite sum andintegrations over a finite interval.

Now assume f ∈ L1(R) ∩ AC(R) as in the hypothesis of PSF. FromLemma 2.5 it follows in particular that f(n) → 0 as n → ±∞. Hence we mayapply (5.6) to f(x) and to f(−x), add the results and let n approach infinity.This way we obtain

∞∑

k=−∞f(k) =

∫

R

f(x) dx− 1π

∫

R

∞∑

k=1

sin(2πkt)k

f ′(t) dt.

Since the partial sums of the series are uniformly bounded and f ∈ AC(R),Lebesgue’s theorem of dominated convergence allows us to interchange inte-gration and summation. Thus,

∞∑

k=−∞f(k) =

∫

R

f(x) dx− 1π

∞∑

k=1

1k

∫

R

sin(2πkt)f ′(t) dt. (5.8)

Now, expressing the sine by exponential functions and performing an integra-tion by parts, we find that

− 1π

∞∑

k=1

1k

∫

R

sin(2πkt)f ′(t) dt

=1

2πi

∞∑

k=1

1k

[∫

R

e−i2πktf ′(t) dt−∫

R

ei2πktf ′(t) dt]]

=∞∑

k=1

[∫

R

e−i2πktf(t) dt+∫

R

ei2πktf(t) dt]

=√

2π∞∑

k=1

[f(k) + f(−k)

].

Substituting this in (5.8), we arrive at PSF. �

6. The Euler–Maclaurin Formula and the ApproximateSampling Formula

Proof of EMSF ⇒ ASF. Let us first consider the particular case that f ∈L2(R) ∩ AC(2)(R) with f ′′ ∈ C(R), and define gt(·) := f(·) sinc(t − ·) witht ∈ R fixed. Now apply (1.1) for r = 1 to gt(·) and gt(−·) and add the twoformulae. This yields for each n ∈ N,


n∑

k=−n

gt(k) =

n∫

−n

gt(u) du +12

[gt(n) + gt(−n)] +112

[g′t(n) − g′

t(−n)]

−∞∑

k=1

n∫

−n


(2πk)2g′′

t (u) du.

Since gt, g′′t ∈ L1(R) ∩ C(R) with limj→±∞ gt(j) = limj→±∞ g′

t(j) = 0, oneobtains for n → ∞ that

∞∑

k=−∞gt(u) du =

∫

R

gt(u) du− limn→∞

∞∑

k=1

n∫

−n


(2πk)2g′′

t (u) du.

Now the infinite series on the right-hand side is uniformly convergent withrespect to n ∈ N and one can interchange the limits with summation to deduce

∞∑

k=−∞gt(k) =

∫

R

gt(u) du−∞∑

k=1

∫

R


(2πk)2g′′

t (u) du

=∫

R

gt(u) du−∞∑

k=−∞k =0

1(2πk)2

∫

R

e−i2πkug′′t (u) du.

Integrating by parts twice yields∞∑

k=−∞gt(k) =

∞∑

k=−∞

∫

R

gt(u)e−i2πku du. (t ∈ R). (6.1)

Using now (2.13) with w = 1, we can rewrite the last integral as∫

R

gt(u)e−i2πku du =∫

R

f(u)e−i2πku sinc(t− u) du = f(t) − (R1f)(t).

This is ASF for w = 1 in that particular instance. The general case now fol-lows as in the proof of PSF ⇒ ASF of Sect. 4 by approximating f by its Fejermeans σρf (cf. (2.16)). �

The proof of the foregoing implication is adapted from [16] and that ofthe converse, to follow, is new.

Proof of ASF ⇒ EMSF. Let f ∈ C(2)[0, n]. Assuming again that f(0) =f(n) = 0, we define φ by (5.1). Then (5.2) and (5.3) hold, and we see that φsatisfies all the hypotheses of ASF. Therefore, ASF applies to φ, and we obtain

φ(t) −∞∑

k=−∞φ(k) sinc(t− k) = (Rφ)(t), (6.2)


where the remainder is now given by

(Rφ)(t) =1√2π

∞∑

k=−∞

(1 − e−i2πkt

)(2k+1)π∫

(2k−1)π

φ(v)eitv dv. (6.3)

From (5.3) it follows that the series in (6.3) converges absolutely and uniformly.Now we integrate both sides of (6.2) over [−ρ, ρ] and let ρ approach

infinity. On the left-hand side we easily see by (2.7) that

limρ→∞

ρ∫

−ρ

[φ(t) −

∞∑

k=−∞φ(k) sinc(t− k)

]dt

=

n∫

0

f(t) dt−n∑

k=0

f(k) limρ→∞

ρ∫

−ρ

sinc(t− k) dt

=

n∫

0

f(t) dt−n∑

k=0

f(k).

The right-hand side needs more care. Using the representation (2.14) ofthe integrals, we may rewrite (6.3) as

(Rφ)(t) =∞∑

k=−∞

(ei2πkt − 1

)n∫

0

f(u)e−i2πku sinc(t− u) du.

Again the series converges absolutely and uniformly. Hence, when we integrateover the finite interval [−ρ, ρ], we may interchange integration and summationand employ Fubini’s theorem. This leads us to

ρ∫

−ρ

(Rφ)(t) dt =∞∑

k=−∞

n∫

0

f(u)ak,ρ(u) du−∞∑

k=−∞

n∫

0

f(u)bρ(u)e−i2πku du,

(6.4)

where

ak,ρ(u) :=

ρ−u∫

−ρ−u

ei2πkt sinc t dt and bρ(u) :=

ρ−u∫

−ρ−u

sinc t dt.

At the end of this proof we will show that the integrals in both series of (6.4)are of order O(k−2) as k → ±∞, uniformly with respect to ρ. Therefore thelimit ρ → ∞ may be taken inside the summation. Since ak,ρ(u) and bρ(u) areuniformly continuous as functions of (u, ρ) on [0, n] × [0,∞), that limit mayeven by taken inside the integration. By (2.7), we have


limρ→∞ ak,ρ(u) = δk,0 and lim

ρ→∞ bρ(u) = 1,

where Kronecker’s delta has been used. Hence

limρ→∞

ρ∫

−ρ

(Rφ)(t) dt = −∞∑

k=−∞k =0

n∫

0

f(u)e−i2πku du.

Altogether we haven∫

0

f(t) dt−n∑

k=0

f(k) = −∞∑

k=1

n∫

0

(ei2πku + e−i2πku

)f(u) du. (6.5)

Integrating by parts twice on the right-hand side and noting (5.5) yieldsEMSF (1.1).

It remains to show that the integrals on the right-hand side of (6.4) are oforder O(k−2) as k → ±∞ uniformly with respect to ρ. Indeed, two integrationsby parts yield

ak,ρ(u) =[ei2πkt

i2πksinc t+

ei2πkt

(2πk)2sinc′ t

]t=ρ−u

t=−ρ−u

− 1(2πk)2

ρ−u∫

−ρ−u

ei2πkt sinc′′ t dt.

This is a decomposition of ak,ρ(u) into five terms. When we multiply byf(u) and integrate over [0, n], we see for the last three terms that they areO(k−2) as k → ±∞, uniformly with respect to ρ. For the first two terms weobserve the same asymptotic behaviour after one integration by parts takingf(u) sinc(±ρ− u) as the term to be differentiated and ei2πk(±ρ−u) as the oneto be integrated. Hence

n∫

0

f(u)ak,ρ(u) du = O(

1k2

)

uniformly with respect to ρ.As regards bρ(u), we note that it is uniformly bounded with respect to ρ

and u by Lemma 2.7 b), and so are its derivatives. Hence two integrations byparts, as in the calculation of φ in (5.2), show that also

n∫

0

f(u)bρ(u)e−i2πku du = O(

1k2

)

uniformly with respect to ρ. This justifies the above procedure and completesthe proof. �


7. The Abel–Plana Formula as a Corollary of Poisson’sSummation Formula and of the Approximate SamplingFormula

Proof of PSF ⇒ APSF. Suppose that f satisfies the hypotheses of APSF. Wemay and will assume that, in addition, f(0) = 0. Indeed, we simply have toreplace f by f−f(0) sinc and observe that APSF is trivial for the sinc function.

For N ∈ N, we now define

fN (x) :=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

f(x) if x ∈ [0, N + 12 ],

2f(N + 12 )(N + 1 − x) if x ∈ (N + 1

2 , N + 1],

0 if x ∈ R\[0, N + 1].

Then fN satisfies the hypothesis of PSF, and so

N∑

k=1

f(k) =√

2π{fN (0) +

∞∑

k=1

[fN (2πk) + fN (−2πk)

]}. (7.1)

Obviously, we may write√

2πfN as ϕN + ψN , where

ϕN (v) :=

N+1/2∫

0

e−ivuf(u) du

and

ψN (v) := 2f(N +

12

) N+1∫

N+1/2

e−ivu(N + 1 − u) du.

An easy calculation shows that

ψN (0) +∞∑

k=1

[ψN (2πk) + ψN (−2πk)

]= 0,

and therefore (7.1) reduces to

N∑

k=1

f(k) =

N+1/2∫

0

f(x) dx+∞∑

k=1

[ϕN (2πk) + ϕN (−2πk)

]. (7.2)


Now, by contour integration in the lower half-plane along the rectangle withvertices at 0, −iT, N + 1

2 − iT, N + 12 we find that

ϕN (v) = −iT∫

0

e−vtf(−it) dt+

N+1/2∫

0

e−v(T+it)f(t− iT ) dt

+ i

T∫

0

e−v(t+iN+i/2)f

(N +

12

− it

)dt.

If v ≥ 2π, then, by (1.2), the second integral approaches zero as T → ∞.Hence

ϕN (2πk) = −i∞∫

0

e−2πktf(−it) + i(−1)k

∞∫

0

e−2πktf

(N +

12

− it

)dt

for k ∈ N. Analogous considerations in the upper half-plane yield

ϕN (−2πk) = i

∞∫

0

e−2πktf(it) dt− i(−1)k

∞∫

0

e−2πktf

(N +

12

+ it

)dt.

Thus (7.2) may be rewritten as

N∑

k=1

f(k) −N+1/2∫

0

f(x) dx

= i

∞∑

k=1

{ ∞∫

0

e−2πkt [f(it) − f(−it)]dt

+ (−1)k+1

∞∫

0

e−2πkt

[f

(N +

12

+ it

)− f

(N +

12

− it

)]dt

}.

Next, we want to show that summation and integration may be inter-changed. For x ∈ {0, N + 1

2}, the hypotheses of APSF guarantee the existenceof constants c1 > 0 and c2 > 0 such that

|f(x+ it) − f(x− it)| ≤ c1t for t ∈ [0, 1]

and

|f(x+ it) − f(x− it)| ≤ c2 e2πt for t ∈ [1,∞).

This yields that for k > 1, we have∞∫

0

e−2πkt|f(x+ it) − f(x− it)| dt ≤ c1(2πk)2

+c2e

−2π(k−1)

2π(k − 1)=: Mk.


Obviously,∑∞

k=2Mk < ∞, and so an interchange of summation and integra-tion is justified by the theorem of Beppo Levi. Thus, we obtain

N∑

k=1

f(k) −N+1/2∫

0

f(x) dx

= i

∞∫

0

f(it) − f(−it)e2πt − 1

dt+ i

∞∫

0

f

(N +

12

+ it

)− f

(N +

12

− it

)

e2πt + 1dt.

By (1.3), the last integral approaches zero as N → ∞, and so we arrive at(1.4) for functions satisfying f(0) = 0. �

The above implication was proved in [49] as an application of a differ-ent version of Poisson’s formula, namely f ∈ AC(R) being replaced by f ∈L1(R) ∩BV (R). The following is new.

Proof of ASF ⇒APSF. Let g ∈ C(2)[0, N + 12 ] such that g(0) = g(N + 1

2 ) = 0.Proceeding as in the proof of (6.5), we can deduce from ASF the analogousequation

N+1/2∫

0

g(t) dt−N∑

k=0

g(k) = −∞∑

k=1

N+1/2∫

0

(ei2πku + e−i2πku

)g(u) du. (7.3)

If f satisfies the hypotheses of APSF, then this equation holds for

g(t) := f(t) − f(0) − t

N + 12

[f

(N +

12

)− f(0)

].

Substituting this in (7.3), we find by a short calculation in which we use theformula

∞∑

m=1

1(2m− 1)2

=π2

8

thatN+1/2∫

0

f(t) dt−N∑

k=0

f(k) +12f(0) = −

∞∑

k=1

[ϕN (2πk) + ϕN (−2πk)

](7.4)

with

ϕN (v) :=

N+1/2∫

0

e−ivuf(u) du.

The series on the right-hand side of (7.4) is exactly that in (7.2). Calculatingit as in the previous proof and using (1.3), we arrive at (1.4). �


As mentioned in the introduction, the converse of the two implicationspresented in this section have not been dealt with. The proofs would probablybe so involved that one could not really claim that one is a corollary of theother.

8. A Proof of the Euler–Maclaurin Formula

Among the four formulae (EMSF, APSF, PSF, ASF) under consideration, itis EMSF which permits the most elementary proof. Variants of the subsequentapproach can be found in many textbooks on Analysis.5 Proving the implica-tions between the expositions does not include proofs of their truth or falsity.Only if the truth value of any one of them is established independently will allof them have the same truth value.

Proof of EMSF. Let f ∈ C(2r)[0, n] and define6 p(x) := 12x

2 − 12x + 1

12 . Twointegrations by parts yield

1∫

0

p(x)f ′′(x) dx = −12[f(0) + f(1)

]+

112[f ′(1) − f ′(0)

]+

1∫

0

f(x) dx.

(8.1)

Applying this formula to f( · + k) for k = 0, . . . , n − 1 and summing up theresults, we obtain

n∫

0

p(x)f ′′(x) dx = −12f(0) −

n−1∑

k=1

f(k) − 12f(n)

+112[f ′(n) − f ′(0)

]+

n∫

0

f(x) dx, (8.2)

where p is the 1-periodic continuation of p from [0, 1] to the whole of R. Sincep is an absolutely continuous even function, its Fourier series is an absolutelyconvergent cosine series that represents p. Therefore

p(x) =∞∑

k=0

ak cos(2πkx) =∞∑

k=0

akei2πkx + e−i2πkx

2,

5 Some authors first establish (5.6) with the trigonometric series replaced by the right-hand side of (5.7). For this, just one integration by parts for a Riemann-Stieltjes integral isrequired; see e.g., [2, pp. 149–150], [31, pp. 506–507].6 The attentive reader will note that 2p(x) is the Bernoulli polynomial B2(x).


where a0 =∫ 1

0p(x) dx = 0 and

ak = 2

1∫

0

p(x) cos(2πkx) dx =2

(2πk)2(k ∈ N).

The last integral has been calculated by using (8.1) for the function f(x) =−(2πk)−2 cos(2πkx). Substituting the Fourier series of p in (8.2) and regroup-ing terms, we obtain EMSF for r = 1. The formula for general r is deducedfrom this special case by repeated integration by parts on the left-hand sideof (8.2) such that p is integrated and f ′′ is differentiated.

We only carry out the case of r = 2. Using the notation (3.1) and therelations (3.4)–(3.5), we find by two integrations by parts:

n∫

0

p(x)f ′′(x) dx =

n∫

0

[L2(ix) + L2(−ix)] f ′′(x) dx

=[1i

[L3(ix) − L3(−ix)] f ′′(x) + [L4(ix) + L4(−ix)] f ′′′(x)]x=n

x=0

−n∫

0

[L4(ix) + L4(−ix)] f (4)(x) dx

= −B4

4![f ′′′(n) − f ′′′(0)] −

∞∑

k=1

n∫

0

ei2πkx + e−i2πkx

(2πk)4f (4)(x) dx.

Substituting this in (8.2), we obtain the formula for r = 2. �

9. Quadrature Formulae for Bandlimited Functions

Suppose that the integrals under consideration exist. Then, by regroupingterms, the three summation formulae EMSF, APSF and PSF can be inter-preted as trapezoidal rules with remainders. In fact, we may write EMSF inthe form

n∫

0

f(x) dx =12f(0) +

n−1∑

k=1

f(k) +12f(n) +R[0,n](f) (9.1)

with remainder

R[0,n](f) = −r∑

k=1

B2k

(2k)!

[f (2k−1)(n) − f (2k−1)(0)

]

+ (−1)r∞∑

k=1

n∫

0

ei2πkt + e−i2πkt

(2πk)rf (2r)(t) dt,


APSF as

∞∫

0

f(x) dx =12f(0) +

∞∑

k=1

f(k) +R[0,∞)(f) (9.2)

with

R[0,∞)(f) = −i∞∫

0


dy (9.3)

and PSF as∫

R

f(x) dx =∑

k∈Z

f(k) +RR(f) (9.4)

with

RR(f) = −√

2π∑

k∈Z\{0}f(2πk).

The three remainders are of a very different nature. If f(v) vanishes for|v| ≥ 2π as it is the case when f belongs to the Bernstein space B1

2π, thenRR(f) = 0. In other words, the trapezoidal rule on R with nodes at the integersis exact for absolutely integrable functions that are bandlimited to (−2π, 2π).

In contrast to this observation, ASF needs f to be bandlimited to (−π, π)in order that its remainder vanishes and the classical sampling formula isobtained. A similar phenomenon occurs for polynomials in connection withGaussian quadrature. While a polynomial of degree n − 1 needs at least nsamples for its reconstruction, the Gaussian quadrature formula with n nodesis exact for polynomials up to degree 2n − 1. Other characterizations of theGaussian quadrature formula also hold analogously for the trapezoidal ruleover R applied to bandlimited functions; see [46], [13, Sect. 2.11.2]. It is there-fore justified to call the trapezoidal rule on R a Gaussian quadrature formulafor bandlimited functions.

What happens to the remainders of the other two formulae (9.1) and(9.2) when we apply them to bandlimited functions? In the case of formula(9.2), this question was studied in [48]. The remainder R[0,∞)(f) does not van-ish unless f is an even function. For practical applications, the representation(9.3) has two disadvantages: It involves values of f on the imaginary axis andit does not provide a sequence of gradual approximations. Therefore in [48,Theorem 1] an expansion of R[0,∞](f) in terms of derivatives of f at 0 hasbeen established. It also connects APSF with EMSF. The result is as follows.


Theorem. Let f be an entire function of exponential type σ less than 2π andsuppose that

∫∞0f(x) dx exists in the sense of Cauchy. Then

∞∫

0

f(x) dx =12f(0) +

∞∑

k=1

f(k) +∞∑

j=1

f (2j−1)(0)B2j

(2j)!. (9.5)

The second series converges absolutely.

If, in addition, f is bounded on the whole real line, then the terms of thesecond series can be estimated explicitly and it turns out that they decreaselike O((σ/(2π))2j) as j → ∞. Thus (9.5) includes a representation of R[0,∞)(f)which yields gradual approximations by truncating the second series.

As a generalization of (9.5), it was shown in [48, Theorem 3], again underthe additional hypothesis of boundedness on R, that

∞∫

0

f(x) dx =∞∑

k=0

f(z + k) +∞∑

j=1

f (j−1)(0)Bj(z)j!

(z ∈ C),

where Bj(z) are the Bernoulli polynomials. Both series converge uniformly oncompact subsets of C. Moreover, the second series converges absolutely.

Connections to (9.1) are easily found by writingn∫

0

f(x) dx =

∞∫

0

[f(x) − f(x+ n)

]dx

and applying (9.2) to the right-hand side.A very general class of quadrature formulae involving derivatives at all

the nodes was established in [47]. It contains a generalization of EMSF [47,Theorem 3] and of APSF [47, Corollary 6]. For another quadrature formulaover semi-infinite intervals see [52, formula (1.13)].

10. Epilogue

In a recent paper [7] the five authors considered, among others, the followingfour formulae of mathematical analysis for bandlimited functions, i.e. for func-tions belonging to the Bernstein spaces Bp

σ, σ > 0, p = 1, 2, (see Sect. 2 forthe definition):

Poisson Summation Formula for Bandlimited Functions (PSFB)

For f ∈ B1σ with σ > 0 we have

∫

R

f(u) du =2πσ

∑

k∈Z

f

(2kπσ

).

This means that in B1σ the trapezoidal rule with step size 2π/σ is exact for

integration over R.


PSFB

CSF

GPF RKF

Figure 2. The implications proved in [7]

Classical Sampling Formula (CSF)


f(z) =∑

k∈Z

f

(kπ

σ

)sinc

(σzπ

− k)

(z ∈ C), (10.1)

the convergence being absolute and uniform in strips of bounded width parallelto the real line, thus in particular, on compact sets.

General Parseval Formula (GPF)

For f, g ∈ B2σ with σ > 0 we have

∫

R

f(u)g(u) du =π

σ

∑

k∈Z

f

(kπ

σ

)g

(kπ

σ

).

Reproducing Kernel Formula (RKF)


f(z) =σ

π

∫

R

f(u) sincσ

π(z − u) du (z ∈ C).

In that paper it was shown that these formulae are equivalent. Moreprecisely, the implications indicated in Fig. 2 were established.

Now in [10] it was shown that the classical sampling formula (CSF)for bandlimited functions is equivalent to the approximate sampling formula(ASF) for not necessarily bandlimited functions, those belonging to F 2. Thusall the assertions of Fig.1 are equivalent to those of Fig. 2.

Figure 3 shows the implications proved in the present paper together withthose established in [7,10].

It should be noted that in [7,10] the Bernstein spaces Bpσ were defined via

functions of exponential type rather than in terms of the Fourier transform, as


PSFB

CSF

GPF RKF

ASF

APSF

PSF

EMSF

Figure 3. The equivalence between the bandlimited andnon-bandlimited assertions

we did above in Sect. 2. The powerful Paley–Wiener theorem, however, guar-antees that these two definitions are nevertheless equivalent. In view of thesedifferent definitions the Paley–Wiener theorem has to be used as a side resultin order to establish CSF ⇔ ASF in Fig. 3.

In earlier papers some of the authors have shown that the four prop-ositions dealing with bandlimited functions are equivalent to Gauß’s sum-mation of the hypergeometric function, the generalised Vandermonde–Chouformula, Tschakalov’s sampling theorem, Dougall’s bilateral sum, to the func-tional equation of the Riemann zeta function and to the transformation for-mula for Jacobi’s elliptic theta function.

For results of this type showing the equivalence of basic formulae in anal-ysis to those in the diagrams above see, e.g. [8,9,12,32–35,55].

11. Short Biography of A. Ostrowski

Alexander (Markowich) Ostrowski (1893–1986) was born in Kiev and was astudent of Chebyshev’s disciple D. A. Grave at the University there; Graverecommended him to E. Landau and K. Hensel. He then studied under Henselat Marburg University from 1912 on, moved to Gottingen in 1918 where hereceived his doctorate under Hilbert and Landau in 1920. The next stop wasHamburg where, as assistant to E. Hecke, he was awarded the Habilitationdegree in 1922. He returned to Gottingen as Dozent, spent the year 1925/1926on a Rockefeller Research Fellowship at Oxford, Cambridge and Edinburgh,and finally was offered the mathematics chair at Basel. He retired in 1958.Ostrowski was the author of some 275 publications, dealing with linear algebra,algebraic equations, estimating their roots, Galois theory, algebraic numbertheory, differential equations, complex analysis, conformal mappings, numeri-cal analysis, Cauchy functional equation, Cauchy–Frullani integrals, methodsfor both finding and approximating eigenvalues of linear systems. In fact, hisworks were published in six volumes [43].


Ostrowski, one of the great mathematicians of the 20th century, was espe-cially known for his desire to unravel the essential features of a problem, for hiselegant and succinct proofs, for investigating a topic exhaustively, once havingselected it, for his thoroughness based on his deep knowledge of the contem-porary literature and especially the original sources. His 16 doctoral studentsincluded S. E. Warschawski (1932), who helped build up the University of Min-nesota’s mathematics department, T. Motzkin (1936), the well-known analyst,E. Batschelet (1944), known for his work in statistics and biomathematics,Werner Gautschi and his twin brother Walter Gautschi (1954), the eminentnumerical analyst. For Ostrowski’s life and work see, in particular, [28].

The paper on the Euler–Maclaurin formula [16] appeared in the volumededicated to Ostrowski’s 90th birthday. He invited PLB to a colloquium lectureat Basel in 1957 and he participated together with his wife Margaret in thesymposium on “Abstract Spaces and Approximation” conducted by PLB, whoalso had the honour to be invited to his villa at Montagnola in the seventies.

Acknowledgements

The authors wish to express their sincere thanks to Walter Gautschi, PurdueUniversity, for informing them of important literature in the early stages oftheir manuscript and for examining the final version carefully. Kind sugges-tions of Pierre de la Harpe, Geneva, led to an improvement of the presentationof the new results in the introduction and those of Rowland Higgins, Montclar,to an improvement of the English. Thanks are due to Barbara Giese, LehrstuhlA fur Mathematik, RWTH Aachen, for preparing the figures.

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P. L. ButzerLehrstuhl A fur MathematikRWTH Aachen University52056 AachenGermanye-mail: [email protected]

P. J. S. G. FerreiraIEETA/DETIUniversidade de Aveiro3810-193 AveiroPortugale-mail: [email protected]

G. SchmeisserDepartment of MathematicsUniversity of Erlangen-NurnbergBismarckstraße 1 1

2

91054 ErlangenGermanye-mail: [email protected]

R. L. StensLehrstuhl A fur MathematikRWTH Aachen University52056 AachenGermanye-mail: [email protected]

Received: November 4, 2010.

Accepted: November 23, 2010.

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